Question: A regular 12-gon is inscribed in a circle of radius 12. The sum of the lengths of all sides and diagonals of the 12-gon can be written in the form
\[a + b \sqrt{2} + c \sqrt{3} + d \sqrt{6},\]where $a$, $b$, $c$, and $d$ are positive integers. Find $a+b+c+d$.
Answer: Let the 12-gon be $ABCDEFGHIJKL,$ and let $O$ be the center, so $OA = 12.$

[asy]
unitsize (3 cm);

pair O = (0,0);
int i, j;

for (i = 0; i <= 11; ++i) {
for (j = i + 1; j <= 11; ++j) {
  draw(dir(30*i)--dir(30*j));
}}

label("$A$", dir(0), dir(0));
label("$B$", dir(30), dir(30));
label("$C$", dir(60), dir(60));
label("$D$", dir(90), dir(90));
label("$E$", dir(120), dir(120));
label("$F$", dir(150), dir(150));
label("$G$", dir(180), dir(180));
label("$H$", dir(210), dir(210));
label("$I$", dir(240), dir(240));
label("$J$", dir(270), dir(270));
label("$K$", dir(300), dir(300));
label("$L$", dir(330), dir(330));
label("$O$", O, NE, UnFill);
[/asy]

Let $P$ be a point such that $OP = 12,$ and let $\theta = \angle AOP.$  Let $Q$ be the midpoint of $\overline{AP}.$

[asy]
unitsize(4 cm);

pair A, O, P, Q;

A = (1,0);
O = (0,0);
P = dir(40);
Q = (A + P)/2;

draw(A--O--P--cycle);
draw(O--Q);

label("$A$", A, E);
label("$O$", O, W);
label("$P$", P, NE);
label("$Q$", Q, E);
label("$12$", (O + A)/2, S);
[/asy]

Then $\angle AOQ = \frac{\theta}{2},$ so $AQ = 12 \sin \frac{\theta}{2},$ and $AP = 24 \sin \frac{\theta}{2}.$

Counting up the sides and diagonals, the sum we want is
\[12AB + 12AC + 12AD + 12AE + 12AF + 6AG.\]We see that $AC = 12,$ $AD = 12 \sqrt{2},$ $AE = 12 \sqrt{3},$ and $AG = 24.$  Also,
\begin{align*}
AB + AF &= 24 \sin 15^\circ + 12 \sin 75^\circ \\
&= 12 \sin 45^\circ \cos 30^\circ \\
&= 12 \cdot \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2} \\
&= 12 \sqrt{6},
\end{align*}so
\begin{align*}
&12AB + 12AC + 12AD + 12AE + 12AF + 6AG \\
&= 12AC + 12AD + 12AE + 12(AB + AF) + 12AG \\
&= 12 \cdot 12 + 12 \cdot 12 \sqrt{2} + 12 \cdot 12 \sqrt{3} + 12 \cdot 12 \sqrt{6} + 6 \cdot 24 \\
&= 288 + 144 \sqrt{2} + 144 \sqrt{3} + 144 \sqrt{6}.
\end{align*}Then $a + b + c + d = 288 + 144 + 144 + 144 = \boxed{720}.$